The unique γ-min labelings of graphs

Abstract

Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f: V (G) → {0, 1, 2, …, m} that induces an edge-labeling f′: E(G) → {1, 2, …, m} on G defined by f′ (e) = |f(u) − f(v)|, for each edge e = uv in E(G). The value of f is defined as val (Formula presented) The maximum value of a γ-labeling of G is defined as valmax(G) = max{val(f): f is a γ-labeling of G}; while the minimum value of a γ-labeling of G is valmin(G) = min{val(f): f is a γ-labeling of G}. A γ-labeling g of G is a γ-max labeling if val(g) = valmax(G) and a γ-labeling h is a γ-min labeling if val(h) = valmin(G). For a γ-labeling f of a graph G of size m, the complementary labeling (Formula presented) of f is defined by (Formula presented). Let G be a connected graph and f a γ-min labeling of G. Then G has a unique γ-min labeling if f and (Formula presented) are only two γ-min labelings of G. In this paper, we study a connected graph having the unique γ-min labeling. The minimum value of a γ-labeling is determined for some classes of trees. Spontaneously, we are able to find that they have no unique γ-min labeling. © 2018 by the Mathematical Association of Thailand. All rights reserved.